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8.2 Some formal considerations

Before we proceed with a more involved example, let us formalize the procedure outlined above. We mainly follow the excellent treatment given in [124Jump To The Next Citation Point], although we restrict ourselves to the cases where š”Æ is a finite regular subalgebra of š”¤.

In the previous example, we performed the decomposition of the roots (and the ensuing decomposition of the algebra) with respect to one of the simple roots which then defined the level. In general, one may consider a similar decomposition of the roots of a rank r Kac–Moody algebra with respect to an arbitrary number s < r of the simple roots and then the level ā„“ is generalized to the “multilevel” ā„“ = (ā„“1,⋅⋅⋅ ,ā„“s).

8.2.1 Gradation

We consider a Kac–Moody algebra š”¤ of rank r and we let š”Æ ⊂ š”¤ be a finite regular rank m < r subalgebra of š”¤ whose Dynkin diagram is obtained by deleting a set of nodes š’© = {n1,⋅⋅⋅ ,ns }(s = r − m ) from the Dynkin diagram of š”¤.

Let γ be a root of š”¤,

∑ ∑ γ = mi αi + ā„“aαa. (8.12 ) iāˆ•∈š’© a∈š’©
To this decomposition of the roots corresponds a decomposition of the algebra, which is called a gradation of š”¤ and which can be written formally as
⊕ š”¤ = š”¤ā„“, (8.13 ) ā„“∈ā„¤s
where for a given ā„“, š”¤ā„“ is the subspace spanned by all the vectors eγ with that definite value ā„“ of the multilevel,
[h, eγ] = γ (h )eγ, la(γ) = ā„“a. (8.14 )
Of course, if š”¤ is finite-dimensional this sum terminates for some finite level, as in Equation (8.11View Equation) for š”°š”©(3,ā„ ). However, in the following we shall mainly be interested in cases where Equation (8.13View Equation) is an infinite sum.

We note for further reference that the following structure is inherited from the gradation:

[š”¤ ā„“,š”¤ā„“′] ⊆ š”¤ ā„“+ā„“′. (8.15 )
This implies that for ā„“ = 0 we have
[š”¤ ,š”¤ ′] ⊆ š”¤ ′, (8.16 ) 0 ā„“ ā„“
which means that š”¤ ā„“′ is a representation of š”¤0 under the adjoint action. Furthermore, š”¤0 is a subalgebra. Now, the algebra š”Æ is a subalgebra of š”¤0 and hence we also have
[š”Æ,š”¤ā„“′] ⊆ š”¤ā„“′, (8.17 )
so that the subspaces š”¤ā„“ at definite values of the multilevel are invariant subspaces under the adjoint action of š”Æ. In other words, the action of š”Æ on š”¤ ā„“ does not change the coefficients ā„“a.

At level zero, ā„“ = (0,⋅⋅⋅ ,0), the representation of the subalgebra š”Æ in the subspace š”¤0 contains the adjoint representation of š”Æ, just as in the case of š”°š”©(3, ā„) discussed in Section 8.1. All positive and negative roots of š”Æ are relevant. Level zero contains in addition s singlets for each of the Cartan generator associated to the set š’©.

Whenever one of the ā„“a’s is positive, all the other ones must be non-negative for the subspace š”¤ā„“ to be nontrivial and only positive roots appear at that value of the multilevel.

8.2.2 Weights of š–Œ and weights of š–—

Let V be the module of a representation ā„›(š”¤) of š”¤ and Λ ∈ š”„ ā‹†š”¤ be one of the weights occurring in the representation. We define the action of h ∈ š”„š”¤ in the representation ā„› (š”¤) on x ∈ V as

h ⋅ x = Λ(h)x (8.18 )
(we consider representations of š”¤ for which one can speak of “weights” [116Jump To The Next Citation Point]). Any representation of š”¤ is also a representation of š”Æ. When restricted to the Cartan subalgebra š”„ š”Æ of š”Æ, Λ defines a weight ¯ ā‹† Λ ∈ š”„š”Æ, which one can realize geometrically as follows.

The dual space š”„ā‹†š”Æ may be viewed as the m-dimensional subspace Π of š”„ ā‹†š”¤ spanned by the simple roots αi, i āˆ•∈ š’©. The metric induced on that subspace is positive definite since š”Æ is finite-dimensional. This implies, since we assume that the metric on š”„ā‹† š”¤ is nondegenerate, that š”„ā‹† š”¤ can be decomposed as the direct sum

ā‹† ā‹† ⊥ š”„š”¤ = š”„š”Æ ⊕ Π . (8.19 )
To that decomposition corresponds the decomposition
āˆ„ ⊥ Λ = Λ + Λ (8.20 )
of any weight, where Λ āˆ„ ∈ š”„ ā‹†≡ Π š”Æ and Λ⊥ ∈ Π ⊥. Now, let h = ∑ kiα∨ ∈ š”„š”Æ i (i∈āˆ•š’©). One has āˆ„ ⊥ āˆ„ Λ (h) = Λ (h) + Λ (h) = Λ (h ) because ⊥ Λ (h) = 0: The component perpendicular to ā‹† š”„š”Æ drops out. Indeed, Λ ⊥(α∨ ) = 2(Λ⊥|αi)-= 0 i (αi|αi) for i∈āˆ•š’©.

It follows that one can identify the weight ā‹† ¯Λ ∈ š”„š”Æ with the orthogonal projection āˆ„ ā‹† Λ ∈ š”„š”Æ of Λ ∈ š”„ā‹†š”¤ on š”„ā‹†š”Æ. This is true, in particular, for the fundamental weights Λi. The fundamental weights Λi project on 0 for i ∈ š’© and project on the fundamental weights ¯Λi of the subalgebra š”Æ for i āˆ•∈ š’©. These are also denoted λ i. For a general weight, one has

Λ = ∑ p Λ + ∑ k Λ (8.21 ) i i a a iāˆ•∈š’© a∈š’©
and
∑ ¯Λ = Λāˆ„ = piλi. (8.22 ) iāˆ•∈š’©
The coefficients pi can easily be extracted by taking the scalar product with the simple roots,
--2---- pi = (α |α )(αi|Λ ), (8.23 ) i i
a formula that reduces to
pi = (αi|Λ ) (8.24 )
in the simply-laced case. Note that āˆ„ āˆ„ (Λ |Λ ) > 0 even when Λ is non-spacelike.

8.2.3 Outer multiplicity

There is an interesting relationship between root multiplicities in the Kac–Moody algebra š”¤ and weight multiplicites of the corresponding š”Æ-weights, which we will explore here.

For finite Lie algebras, the roots always come with multiplicity one. This is in fact true also for the real roots of indefinite Kac–Moody algebras. However, as pointed out in Section 4, the imaginary roots can have arbitrarily large multiplicity. This must therefore be taken into account in the sum (8.13View Equation).

Let ā‹† γ ∈ š”„š”¤ be a root of š”¤. There are two important ingredients:

It follows that the root multiplicity of γ is given as a sum over its multiplicities as a weight in the various representations {ā„› (ā„“q)|q = 1,⋅⋅⋅ ,Nā„“} at level ā„“. Some representations can appear more than once at each level, and it is therefore convenient to introduce a new measure of multiplicity, called the outer multiplicity μ(ā„› (qā„“)), which counts the number of times each representation ā„›(qā„“) appears at level ā„“. So, for each representation at level ā„“ we must count the individual weight multiplicities in that representation and also the number of times this representation occurs. The total multiplicity of γ can then be written as

∑Nā„“ mult(γ) = μ(ā„› (ā„“q))mult ā„›(ā„“)(γ). (8.25 ) q=1 q
This simple formula might provide useful information on which representations of š”Æ are allowed within š”¤ at a given level. For example, if γ is a real root of š”¤, then it has multiplicity one. This means that in the formula (8.25View Equation), only the representations of š”Æ for which γ has weight multiplicity equal to one are permitted. The others have (ā„“) μ (ā„› q ) = 0. Furthermore, only one of the permitted representations does actually occur and it has necessarily outer multiplicity equal to one, (ā„“) μ (ā„› q ) = 1.

The subspaces š”¤ ā„“ can now be written explicitly as

N ⌊μ(ā„›(qā„“)) ⌋ ⊕ ā„“ ⌈ ⊕ [k] (ā„“) ⌉ š”¤ā„“ = ā„’ (Λ q ) , (8.26 ) q=1 k=1
where (ā„“) ā„’(Λ q ) denotes the module of the representation (ā„“) ā„› q and N ā„“ is the number of inequivalent representations at level ā„“. It is understood that if (ā„“) μ(ā„› q ) = 0 for some ā„“ and q, then ā„’ (Λ(qā„“)) is absent from the sum. Note that the superscript [k] labels multiple modules associated to the same representation, e.g., if (ā„“) μ(ā„› q ) = 3 this contributes to the sum with a term
[1] (ā„“) [2] (ā„“) [3] (ā„“) ā„’ (Λ q ) ⊕ ā„’ (Λq ) ⊕ ā„’ (Λ q ). (8.27 )

Finally, we mention that the multiplicity mult (α) of a root ā‹† α ∈ š”„ can be computed recursively using the Peterson recursion relation, defined as [116Jump To The Next Citation Point]

∑ (α|α − 2ρ )c = (γ |γ′)c c ′, (8.28 ) α ′ γ γ γγ+,γ′γ ∈=Qα +
where Q+ denotes the set of all positive integer linear combinations of the simple roots, i.e., the positive part of the root lattice, and ρ is the Weyl vector (defined in Section 4). The coefficients cγ are defined as
∑ 1- ( γ) cγ = k mult k , (8.29 ) k≥1
and, following [19Jump To The Next Citation Point], we call this the co-multiplicity. Note that if γ āˆ•k is not a root, this gives no contribution to the co-multiplicity. Another feature of the co-multiplicity is that even if the multiplicity of some root γ is zero, the associated co-multiplicity cγ does not necessarily vanish. Taking advantage of the fact that all real roots have multiplicity one it is possible, in principle, to compute recursively the multiplicity of any imaginary root. Since no closed formula exists for the outer multiplicity μ, one must take a detour via the Peterson relation and Equation (8.25View Equation) in order to find the outer multiplicity of each representation at a given level. We give in Table 38 a list of root multiplicities and co-multiplicities of roots of AE3 up to height 10.


Table 38: Multiplicities m α = mult(α ) and co-multiplicities cα of all roots α of AE3 up to height 10.
ā„“ m 1 m 2 c α m α α2
0 0 1 1 1 2
0 0 k > 1 1āˆ•k 0 2 2 k
0 1 0 1 1 2
0 k > 1 1 1āˆ•k 0 2 k2
1 0 0 1 1 2
k > 0 0 0 1āˆ•k 0 2 2 k
0 1 1 1 1 2
0 k > 1 k > 1 1āˆ•k 0 2 k2
1 1 0 1 1 0
2 2 0 3/2 1 0
3 3 0 4/3 1 0
4 4 0 7/4 1 0
5 5 0 6/5 1 0
1 1 1 1 1 0
2 2 2 3/2 1 0
3 3 3 4/3 1 0
1 2 0 1 1 2
2 4 0 1/2 0 8
3 6 0 1/3 0 2
2 1 0 1 1 2
4 2 0 1/2 0 8
6 3 0 1/3 0 18
1 2 1 1 1 0
2 4 2 3/2 1 0
2 1 1 1 1 2
4 2 2 1/2 0 8
1 2 2 1 1 2
2 4 4 1/2 0 8
2 2 1 2 2 -2
4 4 2 8 7 -8
2 3 0 1 1 2
4 6 0 1/2 0 8
3 2 0 1 1 2
6 4 0 1/2 0 8
 
ā„“ m 1 m 2 c α m α α2
2 3 1 2 2 -2
3 2 1 1 1 0
2 4 1 1 1 2
2 3 2 2 2 -2
3 2 2 1 1 2
3 3 1 3 3 -4
3 4 0 1 1 2
4 3 0 1 1 2
2 3 3 1 1 2
3 4 1 3 3 -4
2 4 3 1 1 2
3 3 2 3 3 -4
4 3 1 2 2 -2
3 4 2 5 5 -6
3 5 1 1 1 0
4 3 2 2 2 -2
4 4 1 5 5 -6
4 5 0 1 1 2
5 4 0 1 1 2
3 4 3 3 3 -4
3 5 2 3 3 -4
4 3 3 1 1 2
4 5 1 5 5 -6
5 4 1 3 3 -4


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